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Properties

Label 15.15.2034742233...6928.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 37^{5}\cdot 701^{12}\cdot 1061^{12}$
Root discriminant $263{,}370.60$
Ramified primes $2, 37, 701, 1061$
Class number Not computed
Class group Not computed
Galois group $(C_5^2 : C_3):C_2$ (as 15T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51309930300889700898010228382464, -103562164349148797000704434199808, 16384877146005349721822978906880, -75264948123079550954015902464, -8381661447592221588178895680, 5760971201312575426356160, 696538506670849818021440, 1941351085012979462976, -10310407818341917488, -40168727583503152, 53550385971468, 291116427176, -89072821, -892509, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 892509*x^13 - 89072821*x^12 + 291116427176*x^11 + 53550385971468*x^10 - 40168727583503152*x^9 - 10310407818341917488*x^8 + 1941351085012979462976*x^7 + 696538506670849818021440*x^6 + 5760971201312575426356160*x^5 - 8381661447592221588178895680*x^4 - 75264948123079550954015902464*x^3 + 16384877146005349721822978906880*x^2 - 103562164349148797000704434199808*x - 51309930300889700898010228382464)
 
gp: K = bnfinit(x^15 - 3*x^14 - 892509*x^13 - 89072821*x^12 + 291116427176*x^11 + 53550385971468*x^10 - 40168727583503152*x^9 - 10310407818341917488*x^8 + 1941351085012979462976*x^7 + 696538506670849818021440*x^6 + 5760971201312575426356160*x^5 - 8381661447592221588178895680*x^4 - 75264948123079550954015902464*x^3 + 16384877146005349721822978906880*x^2 - 103562164349148797000704434199808*x - 51309930300889700898010228382464, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 892509 x^{13} - 89072821 x^{12} + 291116427176 x^{11} + 53550385971468 x^{10} - 40168727583503152 x^{9} - 10310407818341917488 x^{8} + 1941351085012979462976 x^{7} + 696538506670849818021440 x^{6} + 5760971201312575426356160 x^{5} - 8381661447592221588178895680 x^{4} - 75264948123079550954015902464 x^{3} + 16384877146005349721822978906880 x^{2} - 103562164349148797000704434199808 x - 51309930300889700898010228382464 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2034742233090714111568770651869016171882843590116942007586912219502812217523196928=2^{10}\cdot 37^{5}\cdot 701^{12}\cdot 1061^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $263{,}370.60$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 37, 701, 1061$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{2975044} a^{5} - \frac{1}{2975044} a^{4} - \frac{74376}{743761} a^{3} - \frac{133877}{743761} a^{2} + \frac{236516}{743761} a + \frac{169042}{743761}$, $\frac{1}{5950088} a^{6} - \frac{297505}{5950088} a^{4} - \frac{208253}{1487522} a^{3} - \frac{320561}{743761} a^{2} - \frac{338203}{1487522} a + \frac{84521}{743761}$, $\frac{1}{5950088} a^{7} - \frac{1}{5950088} a^{5} + \frac{178503}{2975044} a^{4} + \frac{413531}{2975044} a^{3} + \frac{607861}{1487522} a^{2} + \frac{185970}{743761} a + \frac{263696}{743761}$, $\frac{1}{23800352} a^{8} + \frac{1}{23800352} a^{6} + \frac{1}{11900176} a^{5} - \frac{596497}{5950088} a^{4} - \frac{145301}{2975044} a^{3} - \frac{927589}{2975044} a^{2} + \frac{469813}{1487522} a - \frac{634897}{1487522}$, $\frac{1}{23800352} a^{9} + \frac{1}{23800352} a^{7} - \frac{1}{11900176} a^{6} - \frac{1}{5950088} a^{5} - \frac{589593}{5950088} a^{4} + \frac{203225}{2975044} a^{3} + \frac{146991}{1487522} a^{2} - \frac{248902}{743761} a - \frac{229251}{743761}$, $\frac{1}{53105320811616} a^{10} + \frac{743759}{53105320811616} a^{9} - \frac{595007}{53105320811616} a^{8} + \frac{89251}{17701773603872} a^{7} - \frac{1701725}{26552660405808} a^{6} + \frac{1691729}{13276330202904} a^{5} + \frac{1134284207311}{13276330202904} a^{4} + \frac{453101337829}{3319082550726} a^{3} + \frac{46763199025}{3319082550726} a^{2} - \frac{79259794274}{553180425121} a + \frac{599428177934}{1659541275363}$, $\frac{1}{106210641623232} a^{11} - \frac{1}{106210641623232} a^{10} + \frac{892513}{106210641623232} a^{9} - \frac{357005}{35403547207744} a^{8} - \frac{452207}{53105320811616} a^{7} - \frac{1762297}{26552660405808} a^{6} - \frac{844615}{13276330202904} a^{5} - \frac{770190185447}{6638165101452} a^{4} - \frac{448550049415}{3319082550726} a^{3} - \frac{44870934415}{2212721700484} a^{2} - \frac{953717077501}{3319082550726} a + \frac{104297100868}{553180425121}$, $\frac{1}{212421283246464} a^{12} - \frac{1}{212421283246464} a^{11} + \frac{1}{212421283246464} a^{10} + \frac{238003}{70807094415488} a^{9} - \frac{2147981}{106210641623232} a^{8} + \frac{4342493}{53105320811616} a^{7} - \frac{2710265}{53105320811616} a^{6} - \frac{1882709}{13276330202904} a^{5} - \frac{1005463595141}{13276330202904} a^{4} - \frac{950702254247}{4425443400968} a^{3} + \frac{2546371148441}{6638165101452} a^{2} + \frac{213843926555}{1106360850242} a - \frac{489769157273}{1106360850242}$, $\frac{1}{424842566492928} a^{13} - \frac{1}{424842566492928} a^{12} + \frac{1}{424842566492928} a^{11} + \frac{1}{424842566492928} a^{10} - \frac{4409017}{212421283246464} a^{9} + \frac{249547}{35403547207744} a^{8} - \frac{3074411}{106210641623232} a^{7} - \frac{874413}{17701773603872} a^{6} + \frac{87768}{553180425121} a^{5} + \frac{518657695063}{26552660405808} a^{4} - \frac{939026955197}{13276330202904} a^{3} + \frac{1230763547669}{3319082550726} a^{2} - \frac{1039083687187}{2212721700484} a + \frac{8930841881}{3319082550726}$, $\frac{1}{166208818832097532859262452521280855178721885816262286270637405168989937501678083771934823306626480786768194612844908692537088} a^{14} - \frac{57110310141804395556490208608148974980712305493220073829847905018005704726355489553199275311826830269516863397}{166208818832097532859262452521280855178721885816262286270637405168989937501678083771934823306626480786768194612844908692537088} a^{13} + \frac{97749998269489275377657614436174738064119087502435183117894585706710818691682386518832101151739150955989311397}{166208818832097532859262452521280855178721885816262286270637405168989937501678083771934823306626480786768194612844908692537088} a^{12} + \frac{1301388493156539887890527837269759794139880493219121269758378940877894532349545223621736048900546715027034001}{18467646536899725873251383613475650575413542868473587363404156129887770833519787085770535922958497865196466068093878743615232} a^{11} + \frac{583978150625494516136480957454238081746890923600846291941504693243554726320938444639557101774646706720620023783}{83104409416048766429631226260640427589360942908131143135318702584494968750839041885967411653313240393384097306422454346268544} a^{10} + \frac{102300423434673885413922524845658411895758485296557503053082509839437155372115940549949046795722733380538326600934291}{5194025588503047901851951641290026724335058931758196445957418911530935546927440117872963228332077524586506081651403396641784} a^{9} - \frac{388155074159862067958221350822098070978818488120288087150754175848558974563475839136968033200213011011706510293058585}{20776102354012191607407806565160106897340235727032785783829675646123742187709760471491852913328310098346024326605613586567136} a^{8} + \frac{965443483452419431768388444944961934667075460488584657041502412408775586153025765226732773267761870201181384728302657}{20776102354012191607407806565160106897340235727032785783829675646123742187709760471491852913328310098346024326605613586567136} a^{7} - \frac{1516966255023931984532224563433696163971757816525288677828276555536158634785550472500597549150111290307784269316312573}{20776102354012191607407806565160106897340235727032785783829675646123742187709760471491852913328310098346024326605613586567136} a^{6} + \frac{80340860619400614705413132620506375774862398410385161061152781134204618099238068001257784977101838819769360095435959}{577113954278116433539105737921114080481673214639799605106379879058992838547493346430329247592453058287389564627933710737976} a^{5} - \frac{312997963319720214478439165882900236350580407293407043193827794762331593072009181794908318413261841891171826884362841634181}{2597012794251523950925975820645013362167529465879098222978709455765467773463720058936481614166038762293253040825701698320892} a^{4} - \frac{92654919734307799076866870245400926158001497587823169406729671142126017051243536167154947295308995420243986223395743279339}{865670931417174650308658606881671120722509821959699407659569818588489257821240019645493871388679587431084346941900566106964} a^{3} + \frac{94027026528825363228779027686183738012270375105831860199632449385039537463136212422859703047458578241023059727847543688391}{865670931417174650308658606881671120722509821959699407659569818588489257821240019645493871388679587431084346941900566106964} a^{2} - \frac{16788601221442503942841399603535992644402833478576972639001485234811232042355905962898103478887411756444250871503964185538}{72139244284764554192388217240139260060209151829974950638297484882374104818436668303791155949056632285923695578491713842247} a + \frac{304667648330600068259667899122038794507806494904970579521089363062435261844165305406581855567150776175930063684938626418355}{1298506397125761975462987910322506681083764732939549111489354727882733886731860029468240807083019381146626520412850849160446}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:S_3$ (as 15T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 13 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

Intermediate fields

3.3.148.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
Degree 30 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$37$37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
701Data not computed
1061Data not computed